Stabilization through self-coupling in networks of small-world and scale-free topology

Mechanisms that ensure the stability of dynamical systems are of vital importance, in particular in our globalized and increasingly interconnected world. The so-called connectivity-stability dilemma denotes the theoretical finding that increased connectivity between the components of a large dynamical system drastically reduces its stability. This result has promoted controversies within ecology and other fields of biology, especially, because organisms as well as ecosystems constitute systems that are both highly connected and stable. Hence, it has been a major challenge to find ways to stabilize complex systems while preserving high connectivity at the same time. Investigating the stability of networks that exhibit small-world or scale-free topology is of particular interest, since these topologies have been found in many different types of real-world networks. Here, we use an approach to stabilize recurrent networks of small-world and scale-free topology by increasing the average self-coupling strength of the units of a network. For both topologies, we find that there is a sharp transition from instability to asymptotic stability. Then, most importantly, we find that the average self-coupling strength needed to stabilize a system increases much slower than its size. It appears that the qualitative shape of this relationship is the same for small-world and scale-free networks, while scale-free networks can require higher magnitudes of self-coupling. We further explore the stabilization of networks with Kronecker-Leskovec topology. Finally, we argue that our findings, in particular the stabilization of large recurrent networks through small increases in the unit self-regulation, are of practical importance for the stabilization of diverse types of complex systems.


Reproduction of results on Erdős-Rényi-type random networks
As a first step to validate our implementation we sought to reproduce the results of Gardner and Ashby (1970). Therefore, we generated graphs with randomly chosen links. Two different algorithms were used for this occasion. The first algorithm distributes links in a sample network such that on average (over samples/trials) a certain connectance value is reached. This is the method used by Gardner and Ashby, corresponding to Erdős-Rényi random graphs. The second algorithm distributes an exact number of links in every sample network, thus leading to exactly the same connectance for every trial. This corresponds to Erdős-Rényi-Gilbert random graphs (cf. Fienberg, 2012). Both algorithms lead to directed graphs whose link strengths are drawn from a uniform random distribution in the interval between −1.0 and +1.0. Self-coupling strengths are drawn from in between −1.0 and −0.1 (also cf. Methods in the main article). It may be noteworthy that May (1972) used normally-distributed random numbers in his theoretical study, albeit referring to the results that Gardner and Ashby had obtained with uniformly-distributed coupling strengths.
The resulting graphs for an average as well as for an exact number of connections are shown in Fig. S1a,b. Considering the general trend, there is no difference between these graphs. There is, however, a difference between the results presented here and those presented by Gardner and Ashby (1970). Their curves for N = 7 and N = 10 drop quite sharply to zero at 40 % and 13 %, respectively, while our results suggest a slower descent. Since we could cross-validate the simulation results obtained from our C++ implementation with a second implementation in Matlab, it seems that the results obtained by Gardner and Ashby might have been biased by the small number of trials that they had been able to use back in 1970.
Another important point is to compare the freely directed graphs considered in Fig. S1a,b and the bidirectional asymmetric graphs considered in Fig. S1c,d. There is a certain deviation at lower connectance values, but the curves converge to the same stability value at connectance 100 %, as one would expect. While the freely directed graphs exhibit slightly increased stability, this is negligible for the purpose of our study.
Please note that for all results in the present study, except those shown in Fig. S1a,c, we used "exact number" (Erdős-Rényi-Gilbert-type) algorithms. Moreover, except for the freely directed graphs used for Fig. S1a,b, we considered bidirectional asymmetric graphs.

Variation of connectance and rewiring number for small-world networks
To explore the stability of small-world networks depending on the connection parameters used to construct the networks, we determined the probability of stability of small-world networks with varied rewiring number R and varied number of nearest-neighbor connections k nn . This enabled us to examine the probability of stability as a function of the connectance C sw , which is given by (cf. Eq. 9 in the main article): Sample graphs are shown in Fig. S2 for different rewiring numbers. The number of data points is restricted by the fact that k nn can only take even integer values. Although the curves shown in Fig. S2 represent regular (for R = 0) or small-world networks (for R > 0), they exhibit the same principal behavior as random networks -the probability of stability declines with increasing connectance. Nevertheless, differences occur across different numbers of rewirings. At low connectance values, all networks are stable. At moderate connectance values, rewiring slightly increases the stability. Eventually, at high connectance values, all networks become unstable. of stability of small-world networks. The probability of stability is plotted against the connectance (here, ≈ knn) for networks of N = 100 nodes. Different curves show data for different numbers R of rewired connections. For each value, 10 5 trials were used. Error bars indicate the standard deviation (mostly too small to be visible) estimated as described in Methods. Self-coupling strengths were drawn from a uniform distribution between −2.2 and −1.3. The construction of small-world networks is described in the Methods section of the main article.
To verify that the considered small-world networks did not exhibit a substantial number of high-degree nodes (which would constitute additional scale-free characteristics -see discussion in the main article), we had to check that the degree distribution of the small-world networks would be sufficiently narrow. The plot in Fig. S3a shows that if there is no rewiring, all nodes have a degree of 5 (4 nearest-neighbor connections plus the self-coupling). With increasing rewiring number R, the distribution broadens and the node degree ranges between 1 and 12 ( Fig. S3b-e), but there is still no substantial number of high-degree nodes (also cf. Fig. 1c in the main article).  Fig. 1a in the main article). Nodes are represented by circles and node degree is indicated by circle size. Arrows represent links between different nodes. Self-coupling of nodes has been masked for better visibility. Supplementary Figure S6: Stabilization parameter for networks with alternative preferential attachment parameters. The stabilization parameter (computed from fits to data points obtained from 1000 trials, cf. Eq. 2 and Fig. S5) is plotted relative to the network size N . Parameters for the preferential attachment algorithm (see Methods): m = 4 and m0 = N/2. The error bars show the error propagated from the fit parameters from which the stabilization parameter is computed. (a) Selfcoupling strengths were drawn from an interval of fixed size and varying position. The stabilization parameter was obtained from the mean self-coupling strength. (b) Self-coupling strengths were drawn from an interval with fixed upper bound and varying lower bound. The stabilization parameter was obtained from the lower bound of the self-coupling strength. See Table 2 in the main article for the fit parameters and their uncertainty. indicates similarity to scale-free characteristics (for which typically 2 γ 3; Barabási, 2014). (b) Sample network of N = 1000 (Watts-Strogatz clustering coefficient: 0.000, average shortest path length: 3.80; also cf. Fig. 1a in the main article). Nodes are represented by circles and node degree is indicated by circle size. Arrows represent links between nodes. Self-coupling of nodes has been masked for better visibility.